Preface to the Second EditionAcknowledgements1 The Rise of Value at Risk1.1 The emergence of financial risk management1.2 Market risk management1.3 Risk management before VaR1.4 Value at riskAppendix 1: Types of Market Risk2 Measures of Financial Risk2.1 The Mean-Variance framework for measuring financial risk2.2 Value at risk2.3 Coherent risk measures2.4 ConclusionsAppendix 1: Probability FunctionsAppendix 2: Regulatory Uses of VaR3 Estimating Market Risk Measures: An Introduction and Overview3.1 Data3.2 Estimating historical simulation VaR3.3 Estimating parametric VaR3.4 Estimating coherent risk measures3.5 Estimating the standard errors of risk measure estimators3.6 OverviewAppendix 1: Preliminary Data AnalysisAppendix 2: Numerical Integration Methods4 Non-parametric Approaches4.1 Compiling historical simulation data4.2 Estimation of historical simulation VaR and ES4.3 Estimating confidence intervals for historical simulation VaR and ES4.4 Weighted historical simulation4.5 Advantages and disadvantages of non-parametric methods4.6 ConclusionsAppendix 1: Estimating Risk Measures with Order StatisticsAppendix 2: The BootstrapAppendix 3: Non-parametric Density EstimationAppendix 4: Principal Components Analysis and Factor Analysis5 Forecasting Volatilities, Covariances and Correlations5.1 Forecasting volatilities5.2 Forecasting covariances and correlations5.3 Forecasting covariance matricesAppendix 1: Modelling Dependence: Correlations and Copulas6 Parametric Approaches (I)6.1 Conditional vs unconditional distributions6.2 Normal VaR and ES6.3 The t-distribution6.4 The lognormal distribution6.5 Miscellaneous parametric approaches6.6 The multivariate normal variance-covariance approach6.7 Non-normal variance-covariance approaches6.8 Handling multivariate return distributions with copulas6.9 ConclusionsAppendix 1: Forecasting longer-term Risk Measures7 Parametric Approaches (II): Extreme Value7.1 Generalised extreme-value theory7.2 The peaks-over-threshold approach: the generalised pareto distribution7.3 Refinements to EV approaches7.4 Conclusions8 Monte Carlo Simulation Methods8.1 Uses of monte carlo simulation8.2 Monte carlo simulation with a single risk factor8.3 Monte carlo simulation with multiple risk factors8.4 Variance-reduction methods8.5 Advantages and disadvantages of monte carlo simulation8.6 Conclusions9 Applications of Stochastic Risk Measurement Methods9.1 Selecting stochastic processes9.2 Dealing with multivariate stochastic processes9.3 Dynamic risks9.4 Fixed-income risks9.5 Credit-related risks9.6 Insurance risks9.7 Measuring pensions risks9.8 Conclusions10 Estimating Options Risk Measures10.1 Analytical and algorithmic solutions m for options VaR10.2 Simulation approaches10.3 Delta-gamma and related approaches10.4 Conclusions11 Incremental and Component Risks11.1 Incremental VaR11.2 Component VaR11.3 Decomposition of coherent risk measures12 Mapping Positions to Risk Factors12.1 Selecting core instruments12.2 Mapping positions and VaR estimation13 Stress Testing13.1 Benefits and difficulties of stress testing13.2 Scenario analysis13.3 Mechanical stress testing13.4 Conclusions14 Estimating Liquidity Risks14.1 Liquidity and liquidity risks14.2 Estimating liquidity-adjusted VaR14.3 Estimating liquidity at risk (LaR)14.4 Estimating liquidity in crises15 Backtesting Market Risk Models15.1 Preliminary data issues15.2 Backtests based on frequency tests15.3 Backtests based on tests of distribution equality15.4 Comparing alternative models15.5 Backtesting with alternative positions and data15.6 Assessing the precision of backtest results15.7 Summary and conclusionsAppendix 1: Testing Whether Two Distributions are Different16 Model Risk16.1 Models and model risk16.2 Sources of model risk16.3 Quantifying model risk16.4 Managing model risk16.5 ConclusionsBibliographyAuthor IndexSubject Index

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Measuring Market Risk

We present a finite difference method for solving parabolic partial integro-differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Levy process or, more generally, a time-inhomogeneous jump-diffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Levy measure. We propose an explicit-implicit finite difference scheme which can be used to price European and barrier options in such models. We study stability and convergence of the scheme proposed and, under additional conditions, provide estimates on the rate of convergence. Numerical tests are performed with smooth and nonsmooth initial conditions.

A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models

http://www.cmap.polytechnique.fr/~rama/pide.pdf

This self-contained, practical, entry-level text integrates the basic principles of applied mathematics, applied probability, and computational science for a clear presentation of stochastic processes and control for jump-diffusions in continuous time. The author covers the important problem of controlling these systems and, through the use of a jump calculus construction, discusses the strong role of discontinuous and nonsmooth properties versus random properties in stochastic systems. The book emphasizes modeling and problem solving and presents sample applications in financial engineering and biomedical modeling. Computational and analytic exercises and examples are included throughout. While classical applied mathematics is used in most of the chapters to set up systematic derivations and essential proofs, the final chapter bridges the gap between the applied and the abstract worlds to give readers an understanding of the more abstract literature on jump-diffusions. An additional 160 pages of online appendices are available on a Web page that supplements the book. Audience This book is written for graduate students in science and engineering who seek to construct models for scientific applications subject to uncertain environments. Mathematical modelers and researchers in applied mathematics, computational science, and engineering will also find it useful, as will practitioners of financial engineering who need fast and efficient solutions to stochastic problems. Contents List of Figures; List of Tables; Preface; Chapter 1. Stochastic Jump and Diffusion Processes: Introduction; Chapter 2. Stochastic Integration for Diffusions; Chapter 3. Stochastic Integration for Jumps; Chapter 4. Stochastic Calculus for Jump-Diffusions: Elementary SDEs; Chapter 5. Stochastic Calculus for General Markov SDEs: Space-Time Poisson, State-Dependent Noise, and Multidimensions; Chapter 6. Stochastic Optimal Control: Stochastic Dynamic Programming; Chapter 7. Kolmogorov Forward and Backward Equations and Their Applications; Chapter 8. Computational Stochastic Control Methods; Chapter 9. Stochastic Simulations; Chapter 10. Applications in Financial Engineering; Chapter 11. Applications in Mathematical Biology and Medicine; Chapter 12. Applied Guide to Abstract Theory of Stochastic Processes; Bibliography; Index; A. Online Appendix: Deterministic Optimal Control; B. Online Appendix: Preliminaries in Probability and Analysis; C. Online Appendix: MATLAB Programs

Applied Stochastic Processes and Control for Jump-Diffusions

An implicit method is developed for the numerical solution of option pricing models where it is assumed that the underlying process is a jump diffusion. This method can be applied to a variety of contingent claim valuations, including American options, various kinds of exotic options, and models with uncertain volatility or transaction costs. Proofs of timestepping stability and convergence of a fixed-point iteration scheme are presented. For typical model parameters, it is shown the error is reduced by two orders of magnitude at each iteration. The correlation integral is computed using a fast Fourier transform method. Numerical tests of convergence for a variety of options are presented.

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Robust numerical methods for contingent claims under jump diffusion processes

The Pricing Equations. Analysis of Finite Difference Methods. Special Issues. Coordinate Transformations. Numerical Examples. Index.

Curt Randall

Papers

Pricing Financial Instruments: The Finite Difference Method